A circular platform is mounted on a frictionless vertical axle. If the radius R = 2m and its moment of inertia about the axle is 200 kg·m², it is initially at rest. A 50 kg man stands on the edge of the platform and begins to walk along the edge at the speed of 1 m/s relative to the ground. The time taken by the man to complete one revolution is:

(A) (3π)/2 s
(B) 2π s
(C) π/2 s
(D) π s

To solve this problem, we can apply the principle of conservation of angular momentum. Since the platform is initially at rest, the total angular momentum of the system (man + platform) is zero.

The angular momentum of the man can be calculated as the product of his moment of inertia and angular velocity. Since the man is moving along the edge of the platform, his moment of inertia can be approximated as the product of his mass and the square of his distance from the axis of rotation (radius).

Let's calculate the angular momentum of the man:
Angular momentum of the man = Moment of inertia of the man × Angular velocity of the man

The moment of inertia of the man can be calculated as:
Moment of inertia of the man = mass of the man × (distance from the axis of rotation)^2

Plugging in the given values:
Mass of the man = 50 kg
Distance from the axis of rotation = radius of the platform = 2 m

Moment of inertia of the man = 50 kg × (2 m)^2 = 200 kg·m^2

The angular momentum of the man is given by:
Angular momentum of the man = 200 kg·m^2 × (1 m/s) = 200 kg·m^2/s

Since the total angular momentum of the system is conserved, the angular momentum of the man must be equal in magnitude but opposite in direction to the angular momentum of the platform.

The moment of inertia of the platform is given as 200 kg·m^2. Let's denote the angular velocity of the platform as ω.

The angular momentum of the platform is given by:
Angular momentum of the platform = Moment of inertia of the platform × Angular velocity of the platform

Angular momentum of the platform = 200 kg·m^2 × ω

Since the total angular momentum is zero, we can write the equation:
Angular momentum of the man + Angular momentum of the platform = 0

200 kg·m^2/s + 200 kg·m^2 × ω = 0

Solving for ω:
ω = -1 m/s

The negative sign indicates that the angular velocity of the platform is in the opposite direction to the man's angular velocity.

The angular velocity of the platform can be related to the linear speed of the man using the formula:
ω = v/r

Where v is the linear speed and r is the radius of the platform.

Plugging in the given values:
-1 m/s = 1 m/s / 2 m

We can see that the negative sign indicates the opposite direction of rotation.

Now, to find the time taken by the man to complete one revolution, we can use the formula:
Time = (2π) / ω

Plugging in the value of ω:
Time = (2π) / (-1 m/s)

Simplifying, we get:
Time = -2π s

The negative sign again indicates the opposite direction of rotation. However, the magnitude of time cannot be negative, so we take the absolute value:

Time = | -2π s | = 2π s

Therefore, the time taken by the man to complete one revolution is 2π seconds.

The answer is B. 2\pi s.